Estimating Shannon Entropy from Recurrence Plots

Recurrence plots were first introduced to quantify the recurrence properties of
chaotic dynamics [1]. A few years later, the recurrence quantification analysis was
introduced to transform graphical representations into statistical analysis [2].
Among the different measures introduced, a Shannon entropy was found to be
correlated with the inverse of the largest Lyapunov exponent. The discrepancy
between this and the usual interpretation of a Shannon entropy has been solved by replacing the probabilities ’s used in the entropy formula
The key point is to replace with the number of diagonal segments of non-recurrent points (made of white dots) divided by the number of recurrent points [3]. Indeed, a white dot representing a non-recurrent point is nothing more than a signature of complexity within the data. With this definition, increases as the bifurcation parameter increases (as shown for the Logistic map in Fig. 1b). There is a one-to-one correspondence between the new definition of and the positive largest Lyapunov exponent.

Fig. 1 : Comparison between the Largest Lyapunov Exponent and the Shannon Entropy.

Estimating the Shannon entropy for the Logistic map

The algorithm provided here computed the Shannon Entropy from Recurrence Plots
for the Logistic map versus parameter . It produced Fig. 1. It is also possible to add noise.

A second Fortran code is provided. It computes the Shannon entropy using a recurrence plot from a data file. you have to specify the number of data point (Npoint). Your data file is expected to have a single column. The code returns

the recurrence rate ;

the "determinism" rate ;

the Shannon entropy.

Be aware that the so-called "determinism" rate does not provide in fact a determinism rate for the simple reason that, for instance, it is equal to 0.82 when estimated from a time series produced by the Logistic map with
, a signal which is 100% deterministic ! So, please, interpret with great care this rate. Note also that, for flows, it is definitely better to compute recurrence plots from a "discrete" time series recorded in the Poincaré section of the attractor than from a "continuous" time series. This is discussed in Ref. [3].